Supernova core collapse?

So at some point nuclear fusion in a star becomes unable to sustain the core against its own gravity pressure, then the core collapses and the surface explodes in supernova explosion. What actually happens when the core collapse that makes the surface explode, and why would the core not explode as well?

Studies suggest core collapse triggers an enormous shock wave that blows off the outer layers of the star. A core fragment is all that remains [typically a neutron star]. This fragment can be 'kicked' at surprising speeds in some cases, probably due to asymmetry in the supernova explosion. While progenitor stars are very massive, the surviving core fragment is typically only a little over one solar mass. The remaining material is released into space.

The Chandrasekhar limit is the mass needed to become a Supernova either a Neutron Star or a Black Hole as a remnant.

Mass = 1.39*(Mass of the Sun)* ( 2.765 × 10^30 kg)

Under that limit a Planetary Nebula will form resulting in a White Dwarf.

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The Chandrasekhar limit only applies to the degenerate core of the star. A star with a core which grows more massive than the Chandrasekhar limit will go supernova and produce a neutron star. A progenitor star whose core will become more massive than 1.4 solar masses is about, as Chronos suggested, 8 solar masses total.

The core need not reach the Chandrasekhar limit to become a core collapse candidate, that is merely an upper limit. It need only be massive enough to initiate carbon fusion. Most neutron stars are below the Chandrasekhar mass limit. A few behemoths do, however, exist. They remain a scientific curiosity.

So at some point nuclear fusion in a star becomes unable to sustain the core against its own gravity pressure, then the core collapses and the surface explodes in supernova explosion. What actually happens when the core collapse that makes the surface explode, and why would the core not explode as well?

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I recommend a marvelous article by Hans Bethe and Gerald Brown called How A Supernova Explodes. It will help you understand what happens both in the core and in the surrounding exterior. Great reading.